We study the time complexity of computing the (min⁡,+) matrix product of two n×n integer matrices in terms of n and the number of monotone subsequences the rows of the first matrix and the columns of the second matrix can be decomposed into. In particular, we show that if each row of the first matrix can be decomposed into at most m1 monotone subsequences and each column of the second matrix can be decomposed into at most m2 monotone subsequences such that all the subsequences are non-decreasing or all of them are non-increasing then the (min⁡,+) product of the matrices can be computed in O(m1m2n2.569) time. On the other hand, we observe that if all the rows of the first matrix are non-decreasing and all columns of the second matrix are non-increasing or vice versa then this case is as hard as the general one. We also present six cases of the restrictions on the input integer matrices under which the problem of computing the (min⁡,+) matrix product is equally hard as that of computing the minimum and maximum witnesses of Boolean matrix product. Similarly, we also study the time complexity of computing the (min⁡,+) convolution of two n-dimensional integer vectors in terms of n and the number of monotone subsequences the two vectors can be decomposed into. We show that if the first vector can be decomposed into at most m1 monotone subsequences and the second vector can be decomposed into at most m2 subsequences such that all the subsequences of the first vector are non-decreasing and all the subsequences of the second vector are non-increasing or vice versa then their (min⁡,+) convolution can be computed in O˜(m1m2n1.5) time. On the other, the case when both sequences of consecutive coordinates of the vectors are non-decreasing or both of them are non-increasing is as hard as the general case. Finally, we present six cases of the restrictions on the input integer vectors under which the problem of computing the (min⁡,+) vector convolution is equally hard as that of computing the minimum and maximum witnesses of the Boolean vector convolution.

We study the time complexity of computing the (min, + ) matrix product of two n× n integer matrices in terms of n and the number of monotone subsequences the rows of the first matrix and the columns of the second matrix can be decomposed into. In particular, we show that if each row of the first matrix can be decomposed into at most m1 monotone subsequences and each column of the second matrix can be decomposed into at most m2 monotone subsequences such that all the subsequences are non-decreasing or all of them are non-increasing then the (min, + ) product of the matrices can be computed in O(m1m2n2.569) time. On the other hand, we observe that if all the rows of the first matrix are non-decreasing and all columns of the second matrix are non-increasing or vice versa then this case is as hard as the general one. Similarly, we also study the time complexity of computing the (min, + ) convolution of two n-dimensional integer vectors in terms of n and the number of monotone subsequences the two vectors can be decomposed into. We show that if the first vector can be decomposed into at most m1 monotone subsequences and the second vector can be decomposed into at most m2 subsequences such that all the subsequences of the first vector are non-decreasing and all the subsequences of the second vector are non-increasing or vice versa then their (min, + ) convolution can be computed in O~ (m1m2n1.5) time. On the other, the case when both vectors are non-decreasing or both of them are non-increasing is as hard as the general case.

First, we present a new algorithm for the single-source shortest paths problem (SSSP) in edge-weighted directed graphs, with n vertices, m edges, and both positive and negative real edge weights. Given a positive integer parameter t, in O(tm) time the algorithm finds for each vertex v a path distance from the source to v not exceeding that yielded by the shortest path from the source to v among the so called t+ light paths. A directed path between two vertices is t+ light if it contains at most t more edges than the minimum edge-cardinality directed path between these vertices. For t= O(n), our algorithm yields an O(nm)-time solution to SSSP in directed graphs with real edge weights matching that of Bellman and Ford. Our main contribution is a new, output-sensitive algorithm for the all-pairs shortest paths problem (APSP) in directed acyclic graphs (DAGs) with positive and negative real edge weights. The running time of the algorithm depends on such parameters as the number of leaves in (lexicographically first) shortest-paths trees, and the in-degrees in the input graph. If the trees are sufficiently thin on the average, the algorithm is substantially faster than the best known algorithm. Finally, we discuss an extension of hypothetical improved upper time-bounds for APSP in non-negatively edge-weighted DAGs to include directed graphs with a polynomial number of large directed cycles. © 2022, Springer Nature Switzerland AG.

Henzinger et al. posed the so-called Online Boolean Matrix-vector Multiplication (OMv) conjecture and showed that it implies tight hardness results for several basic dynamic or partially dynamic problems [STOC'15]. We first show that the OMv conjecture is implied by a simple off-line conjecture that we call the MvP conjecture. We then show that if the definition of the OMv conjecture is generalized to allow individual (i.e., it might be different for different matrices) polynomial-time preprocessing of the input matrix, then we obtain another conjecture (called the OMvP conjecture) that is in fact equivalent to our MvP conjecture. On the other hand, we demonstrate that the OMv conjecture does not hold in restricted cases where the rows of the matrix or the input vectors are clustered, and develop new efficient randomized algorithms for such cases. Finally, we present applications of our algorithms to answering graph queries. (c) 2021 Elsevier Inc. All rights reserved.

We study the problem of determining the Boolean product of two n x n Boolean matrices in an unconventional computational model allowing for mechanical operations. We show that O(n(2)) operations are sufficient to compute the product in this model.

We study several problems of clearing subgraphs by mobile agents in digraphs. The agents can move only along directed walks of a digraph and, depending on the variant, their initial positions may be pre-specified. In general, for a given subset S of vertices of a digraph D and a positive integer k, the objective is to determine whether there is a subgraph H = (V-H, A(H)) of D such that (a) S subset of V-H, (b) H is the union of k directed walks in D, and (c) the underlying graph of H includes a Steiner tree for S in D. Since a directed walk is a not necessarily a simple directed path, the problem is actually on covering with paths. We provide several results on the polynomial time tractability, hardness, and parameterized complexity of the problem. Our main fixed-parameter algorithm is randomized. (C) 2018 Elsevier Inc. All rights reserved.

Henzinger et al. posed the so called Online Boolean Matrix-vector Multiplication (OMv) conjecture and showed that it implies tight hardness results for several basic partially dynamic or dynamic problems [STOC’15].

We show that the OMv conjecture is implied by a simple off-line conjecture. If a not uniform (i.e., it might be different for different matrices) polynomial-time preprocessing of the matrix in the OMv conjecture is allowed then we can show such a variant of the OMv conjecture to be equivalent to our off-line conjecture. On the other hand, we show that the OMV conjecture does not hold in the restricted cases when the rows of the matrix or the input vectors are clustered.

We study the problem of computing the so called minimum and maximum witnesses for Boolean vector convolution. We also consider a generalization of the problem which is to determine for each positive value at a coordinate of the convolution vector, q smallest (largest) witnesses, where q is the minimum of a parameter k and the number of witnesses for this coordinate. We term this problem the smallest k-witness problem or the largest k-witness problem, respectively. We also study the corresponding smallest and largest k-witness problems for Boolean matrix product. First, we present an (O) over tilde (n(1.5)k(0.5))-time algorithm for the smallest or largest k-witness problem for the Boolean convolution of two n-dimensional vectors, where the notation (O) over tilde ( ) suppresses polylogarithmic in n factors. In consequence, we obtain new upper time bounds on reporting positions of mismatches in potential string alignments and on computing restricted cases of the (min, +) vector convolution. Next, we present a fast (substantially subcubic in n and linear in k) algorithm for the smallest or largest k-witness problem for the Boolean matrix product of two n x n Boolean matrices. It yields fast algorithms for reporting k lightest (heaviest) triangles in a vertex-weighted graph.

We study the complexity of the so called semi-disjoint bilinear forms over different semi-rings, in particular the n-dimensional vector convolution and n x n matrix product. We consider a powerful unit-cost computational model over the ring of integers allowing for several additional operations and generation of large integers. We show the following dichotomy for such a powerful model: while almost all arithmetic semi-disjoint bilinear forms have the same asymptotic time complexity as that yielded by naive algorithms, matrix multiplication, the so called distance matrix product, and vector convolution can be solved in a linear number of steps. It follows in particular that in order to obtain a non-trivial lower bounds for these three basic problems one has to assume restrictions on the set of allowed operations and/or the size of used integers.

We study several problems of clearing subgraphs by mobile agents in digraphs. The agents can move only along directed walks of a digraph and, depending on the variant, their initial positions may be pre-specified. In general, for a given subset � of vertices of a digraph D and a positive integer k, the objective is to determine whether there is a subgraph �=(��,��) of D such that (a) �⊆��, (b) H is the union of k directed walks in D, and (c) the underlying graph of H includes a Steiner tree for �. We provide several results on parameterized complexity and hardness of the problems.